# Degeneracy factor partition function

With an accout for my. In statistical mechanicsthe partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium.

It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the thermodynamic variables of the system, such as the total energy, free energyentropy, and pressurecan be expressed in terms of the partition function or its derivatives.

There are actually several different types of partition functions, each corresponding to different types of statistical ensemble or, equivalently, different types of free energy. The canonical partition function applies to a canonical ensemblein which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles.

The grand canonical partition function applies to a grand canonical ensemblein which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances. Suppose we have a thermodynamically large system that is in constant thermal contact with the environment, which has temperature Twith both the volume of the system and the number of constituent particles fixed.

This kind of system is called a canonical ensemble. Generally, these microstates can be regarded as discrete quantum states of the system. Sometimes degeneracy of states is also used and the partition function becomes. In classical statistical mechanics, it is not really correct to express the partition function as a sum of discrete terms, as we have done. In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable.

In this case, some form of coarse graining procedure must be carried out, which essentially amounts to treating two mechanical states as the same microstate if the differences in their position and momentum variables are "not too large".

The partition function then takes the form of an integral. For instance, the partition function of a gas of N classical particles is.

The reason for the N! For simplicity, we will use the discrete form of the partition function in this article, but our results will apply equally well to the continuous form. For more details on the derivation of the above classical partition function, see Configuration integral statistical mechanicswhere the partition function is denoted byinstead ofwhich is used to denote a different quantity called configuration integral.

In quantum mechanics, the partition function can be more formally written as a trace over the state space which is independent of the choice of basis :. The exponential of an operator can be defined, for purely physical considerations, using the exponential power series.

It may not be obvious why the partition function, as we have defined it above, is an important quantity. Firstly, let us consider what goes into it. The partition function is a function of the temperature T and the microstate energies E 1E 2E 3etc.

### Partition function (statistical mechanics)

The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles.

This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.

The partition function can be related to thermodynamic properties because it has a very important statistical meaning.The partition function of a system, Q, provides the tools to calculate the probability of a system occupying state i. Partition function depends on composition,volume and number of particle. Larger the partition function allows to have more accessible energy states at that temperature.

The general form of a partition function is a sum over the states of the system. Two equivalent ways could be used to write the the partition function. Sum over states approach allows to give different indices to the states with the same energy.

Energy levels approach suggests that only energy levels with distinct energies have their own index. This requires that the energies levels of the entire system must be known and the calculations have to be calculated from sums over states.

This limits the types of the systems that we can derive properties for. For ideal gases, we assume that the energy states of molecules are independent of those in other molecules. The molecular partition function, q, is defined as the sum over the states of an individual molecule.

The particles of an ideal gas could be considered distinguishable if they are different from each other and, therefore,the unique label could be assigned to each.

On the contrary,the indistinguishable particles are impossible to assign a unique label,as they are the identical to each other. This article considers the indistinguishable particles of an ideal gas for which the partition function of the system Q could be expressed in terms of the molecular partition function q ,and the number of particles in the system N.

As the partition function allows to calculate the probability of the system occupying the state j. Such system could be assumed as isolated,as a microcanonical ensemble of particles,where the total volume,the total energy and number of particles are constant. However,the range of energies appropriate for composition,volume and temperature of canonical ensembles contributing to the system must be still considered for composition,volume and temperature.

The energy levels of a molecule can be approximated as the sum of energies in the various degrees of freedom of the molecule. The translational partition function, q transis the sum of all possible translational energy states, which could be represented using one,two and three dimensional models for a particle in the box equation, depending on the system of the coordinates. The one and two dimensionsal spaces for a particle in the box equation forms are less commonly used than the three dimensional form as those do not account for the force acting on the particles inside of a box.

For a molecule in the three dimensional space, the energy term in the general partition function equation is replaced with the particle in a 3D box equation. All molecules have three translational degrees of freedom, one for each axis the molecule can move along in three dimensional space. The assumption that energy levels are continuous is valid because the space between energy levels is extremely small, resulting in minimal error. This form is convenient as it does not include a sum to infinity and can therefore be solved with relative ease.

This sum is found by substituting the equation for the energy levels of a linear rigid rotor:.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. I've been trying to look around the internet for an explanation of it but I can't find one.

I guess it is the number of degenerate states in a given energy level? How do you determine how many degenerate states there are? Thanks for the help! In this case, we say that their corresponding shared energy eigenvalue is degenerate.

As for a simple example, consider a system consisting of two, noninteracting one-dimensional quantum harmonic oscillators.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How do you determine the value of the degeneracy factor in the partition function? Ask Question. Asked 7 years ago. Active 7 years ago. Viewed 10k times. Spaderdabomb Spaderdabomb 1, 1 1 gold badge 9 9 silver badges 17 17 bronze badges.

Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue.

In classical mechanicsthis can be understood in terms of different possible trajectories corresponding to the same energy. Degeneracy plays a fundamental role in quantum statistical mechanics. For an N -particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.

The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert spacewhile the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basisthe components of these vectors and the matrix elements of the operators in that basis may be determined.

The dimension of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracywhich can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement.

## Partition function (statistical mechanics)

The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue.

In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems. In several cases, analytic results can be obtained more easily in the study of one-dimensional systems.

It can be proven that in one dimension, there are no degenerate bound states for normalizable wave functions. Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions.

Real two-dimensional materials are made of monoatomic layers on the surface of solids. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillatorwhich act as useful mathematical models for several real world systems.

Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For two commuting observables A and Bone can construct an orthonormal basis of the state space with eigenvectors common to the two operators. If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian.

These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system. The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Mathematically, the relation of degeneracy with symmetry can be clarified as follows.

Consider a symmetry operation associated with a unitary operator S. If the Hamiltonian remains unchanged under the transformation operation Swe have. The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian.

The commutators of the generators of this group determine the algebra of the group.In physicsa partition function describes the statistical properties of a system in thermodynamic equilibrium. Most of the aggregate thermodynamic variables of the system, such as the total energyfree energyentropyand pressurecan be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number. Each partition function is constructed to represent a particular statistical ensemble which, in turn, corresponds to a particular free energy. The most common statistical ensembles have named partition functions.

The canonical partition function applies to a canonical ensemblein which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemblein which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.

Other types of partition functions can be defined for different circumstances; see partition function mathematics for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance. Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature Tand both the volume of the system and the number of constituent particles are fixed.

A collection of this kind of systems comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanicsand whether the spectrum of states is discrete or continuous. For a canonical ensemble that is classical and discrete, the canonical partition function is defined as.

There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach. According to the second law of thermodynamicsa system assumes a configuration of maximum entropy at thermodynamic equilibrium [ citation needed ]. The probabilities of all states add to unity second axiom of probability :.

In the canonical ensemblethe average energy is fixed conservation of energy :. In classical mechanicsthe position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms.

In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as. To make it into a dimensionless quantity, we must divide it by hwhich is some quantity with units of action usually taken to be Planck's constant.In physicsa partition function describes the statistical properties of a system in thermodynamic equilibrium.

Most of the aggregate thermodynamic variables of the system, such as the total energyfree energyentropyand pressurecan be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number. Each partition function is constructed to represent a particular statistical ensemble which, in turn, corresponds to a particular free energy. The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemblein which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles.

The grand canonical partition function applies to a grand canonical ensemblein which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function mathematics for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.

Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature Tand both the volume of the system and the number of constituent particles are fixed. A collection of this kind of systems comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanicsand whether the spectrum of states is discrete or continuous.

For a canonical ensemble that is classical and discrete, the canonical partition function is defined as. There are multiple approaches to deriving the partition function. The following derivation follows the more powerful and general information-theoretic Jaynesian maximum entropy approach. According to the second law of thermodynamicsa system assumes a configuration of maximum entropy at thermodynamic equilibrium [ citation needed ].

The probabilities of all states add to unity second axiom of probability :. In the canonical ensemblethe average energy is fixed conservation of energy :. In classical mechanicsthe position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms.

In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as.

To make it into a dimensionless quantity, we must divide it by hwhich is some quantity with units of action usually taken to be Planck's constant. The reason for the factorial factor N! The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h 3 N where h is usually taken to be Planck's constant.

## Statistical Thermodynamics and Rate Theories/Molecular partition functions

For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:. For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as. In systems with multiple quantum states s sharing the same energy E sit is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels indexed by j as follows:.

The above treatment applies to quantum statistical mechanicswhere a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above. In quantum mechanics, the partition function can be more formally written as a trace over the state space which is independent of the choice of basis :.

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Boltzmann distribution, partition function, and internal energy L2 4449

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